$v$ is a high dimentional point in a vector space.

$$v=\left[\begin{array}{c}x\\ y\end{array}\right]$$

Vectors are written column wise. We can make it row wise by transposing it.

$$v^T=[x_0,x_1,...,x_D]$$

A vector isn't just a list of numbers. We need to have an origin and an orientation of the axes.

Imagine a plane with a looot of dimensions, then a vector is just a point inside of it.
Each component i of the vector is a scalar that stands for the coordinate of the dimension i.

Geometric Interpretation:

Point in space:

Given a vector, the first interpretation that we should give it is as a point in space

"In two or three dimensions, we can visualize these points by using the components of the vectors to define the location of the points in space compared to a fixed reference called the origin.

Direction in space:

The bright side of this one is that the vector addition actually makes sense.
It would just be to follow the directions of the first vector, than we follow the directions from the second vector, getting to the point.

Span:

The span of a vector is the set of all their linear combinations(sum of vectors).
The span of most 2D vectors is a 2D space.

Linear indipendence/Dimensions collapsing

If two basis vectors line up in the same directions, the span lies on a single line.
Another definition, is that if a basis vector sits on the span of another vector, it is redundant and doesn't add anything to the span of the system.

Two vectors are linearly dependent if one is the scaled version of the other.

Inner product (Dot product)

Let $x$ and $y$ be two 2D vectors with their magnitudes $|x|$ and $|y|$.

Now, we take $y$ and compute the length of its projection onto $x$, which we call $y_p$:

$$y_p=|y|\cos \theta$$

Once we have the length of the projection, the dot product is:

$$x\cdot y=|x||y|\cos \theta$$

Or:

There is another way:
$$\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]\cdot \left[\begin{array}{c}{y}_1\\ y_2\\ y_3\end{array}\right]=x_1{y}_1+x_2{y}_2+x_3{y}_4$$

Why are those two ways the same thing?

BECAUSE, imagine if we use a tasformation with only one dimension for 2D vectors.

$$t=\left[\begin{array}{cc}2& 1\end{array}\right]$$

Now, with this transformation, the span of the system becomes a line, because both unit vectors are on the same dimension.

So if we put any vector through this transformation, it will go along this line.

Since the components of our vector are adding value to the same dimension, for us to obtain our output vector, we just need to sum its components:

So the new vector/scalar is -2.
And we can see that applying the transformation to the vector is the same operation that we described before.
In the first way, we are taking the norm/magnitudes of each vector and comparing them in one dimension.
In the second way we are collapsing everything in one single dimension(the one of the first vector), then the other vector will automatically project onto the dimension.

TO BE REVISED!
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Outer product: